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On the multiple time-scales perturbation method for differential-delay equations

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Abstract

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential-delay equations at the unperturbed level (and not from solutions of ordinary differential equations at the unperturbed level as in the classical approach in the literature). This implies that infinitely many roots of the characteristic equation for the unperturbed differential-delay equation are taken into account and that the approximations satisfy initial conditions which are given on a time-interval (determined by the delay). Simple and more advanced examples are treated in detail to show how the method based on differential and difference operators can be applied.

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References

  1. Anacleto, M., Vidal, C.: Dynamics of a delayed predator-prey model with allee effect and holling type II functional response. Math. Methods Appl. Sci. 43(9), 5708–5728 (2020). https://doi.org/10.1002/mma.6307

    Article  MathSciNet  Google Scholar 

  2. Arditi, R., Abillon, J.M., da Silva, J.V.: The effect of a time-delay in a predator-prey model. Math. Biosci. 33(1–2), 107–120 (1977). https://doi.org/10.1016/0025-5564(77)90066-9

    Article  Google Scholar 

  3. Atay, F.M.: Van der pol’s oscillator under delayed feedback. J. Sound Vib. 218(2), 333–339 (1998). https://doi.org/10.1006/jsvi.1998.1843

    Article  MathSciNet  Google Scholar 

  4. Bellman, R., Cook, K.L.: Mathematics in science and engineering. A series of Monographs and Textbooks. Differential-Difference Equations, vol. 6, 1st edn. Academic Press Inc (1963). https://www.sciencedirect.com/bookseries/mathematics-in-science-and-engineering/vol/6/suppl/C

  5. Bender, C.M., Orszag, S.A.: Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media (2013)

  6. Çelik, C.: The stability and hopf bifurcation for a predator-prey system with time delay. Chaos, Solitons Fractals 37(1), 87–99 (2008). https://doi.org/10.1016/j.chaos.2007.10.045

    Article  MathSciNet  Google Scholar 

  7. Das, S.L., Chatterjee, A.: Multiple scales without center manifold reductions for delay differential equations near hopf bifurcations. Nonlinear Dyn. 30(4), 323–335 (2002). https://doi.org/10.1023/a:1021220117746

    Article  MathSciNet  Google Scholar 

  8. Erneux, T.: Multiple time scale analysis of delay differential equations modeling mechanical systems. In: Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C. ASMEDC (2005). https://doi.org/10.1115/detc2005-85028

  9. Ghouli, Z., Hamdi, M., Belhaq, M.: The delayed van der pol oscillator and energy harvesting. In: Springer Proceedings in Physics, pp. 89–109. Springer Singapore (2019). https://doi.org/10.1007/978-981-13-9463-84

  10. Glass, D.S., Jin, X., Riedel-Kruse, I.H.: Nonlinear delay differential equations and their application to modeling biological network motifs. Nat. Commun. (2021). https://doi.org/10.1038/s41467-021-21700-8

    Article  Google Scholar 

  11. Hamdi, M., Belhaq, M.: Quasi-periodic vibrations in a delayed van der pol oscillator with time-periodic delay amplitude. J. Vib. Control 24(24), 5726–5734 (2015). https://doi.org/10.1177/1077546315597821

    Article  MathSciNet  Google Scholar 

  12. Haque, M., Sarwardi, S., Preston, S., Venturino, E.: Effect of delay in a lotka–volterra type predator–prey model with a transmissible disease in the predator species. Math. Biosci. 234(1), 47–57 (2011). https://doi.org/10.1016/j.mbs.2011.06.009

    Article  MathSciNet  Google Scholar 

  13. Holmes, M.H.: Introduction to Perturbation Methods. Springer New York (2013). https://doi.org/10.1007/978-1-4614-5477-9

  14. Hoppensteadt, F.C., Miranker, W.L.: Multitime methods for systems of difference equations. Stud. Appl. Math. 56(3), 273–289 (1977). https://doi.org/10.1002/sapm1977563273

    Article  MathSciNet  Google Scholar 

  15. Huang, C.: Multiple scales scheme for bifurcation in a delayed extended van der pol oscillator. Physica A 490, 643–652 (2018). https://doi.org/10.1016/j.physa.2017.08.035

    Article  MathSciNet  Google Scholar 

  16. Insperger, T., Stépán, G.: Stability chart for the delayed mathieu equation. Proc. Royal Soc. London Ser. A Math. Phys. Eng. Sci. 458(2024), 1989–1998 (2002). https://doi.org/10.1098/rspa.2001.0941

    Article  MathSciNet  Google Scholar 

  17. Insperger, T., Stépán, G.: Stability of the damped mathieu equation with time delay. J. Dyn. Syst. Meas. Contr. 125(2), 166–171 (2003). https://doi.org/10.1115/1.1567314

    Article  Google Scholar 

  18. Kalmár-Nagy, T.: Stability analysis of delay-differential equations by the method of steps and inverse laplace transform. Differ. Eq. Dynam. Syst. 17(1–2), 185–200 (2009). https://doi.org/10.1007/s12591-009-0014-x

    Article  MathSciNet  Google Scholar 

  19. Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer New York (1996). https://doi.org/10.1007/978-1-4612-3968-0

  20. Morrison, T.M., Rand, R.H.: 2:1 resonance in the delayed nonlinear mathieu equation. Nonlinear Dyn. 50(1–2), 341–352 (2007). https://doi.org/10.1007/s11071-006-9162-5

    Article  MathSciNet  Google Scholar 

  21. Murdock, J., Wang, L.C.: Validity of the multiple scale method for very long intervals. ZAMP Zeitschrift fur angewandte Mathematik und Physik 47(5), 760–789 (1996). https://doi.org/10.1007/bf00915274

    Article  MathSciNet  Google Scholar 

  22. Murdock, J.A.: Perturbations: theory and methods. SIAM (1999)

  23. Nandakumar, K., Wahi, P., Chatterjee, A.: Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations. Nonlinear Dyn. 62(4), 705–716 (2010). https://doi.org/10.1007/s11071-010-9755-x

    Article  MathSciNet  Google Scholar 

  24. Nayfeh, A.: Perturbation methods. Physics textbook. Wiley (2008). https://books.google.co.id/books?id=eh6RmWZ51NIC

  25. Nayfeh, A.H.: Introduction to perturbation techniques. John Wiley & Sons (2011)

  26. Nelson, P.: Dynamical Systems Theory, Delay Differential Equations, pp. 637–641. Springer New York, New York, NY (2013)

  27. Perko, L.M.: Higher order averaging and related methods for perturbed periodic and quasi-periodic systems. SIAM J. Appl. Math. 17(4), 698–724 (1969)

    Article  MathSciNet  Google Scholar 

  28. Poincare, H.: New Methods of Celestial Mechanics, NASA technical translations, F-450. I and II, National Aeronautics and Space Administration (1959)

    Google Scholar 

  29. Pontryagin, L.S.: On the Zeros of Some Transcendental Functions (1955). https://doi.org/10.1090/trans2/001/06

  30. Sah, S.M., Rand, R.H.: Three ways of treating a linear delay differential equation. In: Springer Proceedings in Physics, pp. 251–257. Springer International Publishing (2018). https://doi.org/10.1007/978-3-319-63937-6_14

  31. Saha, A., Bhattacharya, B., Wahi, P.: A comparative study on the control of friction-driven oscillations by time-delayed feedback. Nonlinear Dyn. 60(1–2), 15–37 (2009). https://doi.org/10.1007/s11071-009-9577-x

    Article  Google Scholar 

  32. Saha, T., Bandyopadhyay, M.: Multiple scale analysis of a delayed predator prey model within random environment. J Appl. Math. Inform. 26(56), 1191–1205 (2008)

    Google Scholar 

  33. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer (2010). https://doi.org/10.1007/978-1-4419-7646-8

  34. Van Horssen, W.T., Ter Brake, M.: On the multiple scales perturbation method for difference equations. Nonlinear Dyn. 55(4), 401–418 (2009). https://doi.org/10.1007/s11071-008-9373-z

    Article  MathSciNet  Google Scholar 

  35. Verhulst, F.: Methods and Applications of Singular Perturbations. Springer New York (2005). https://doi.org/10.1007/0-387-28313-7

  36. Wahi, P., Chatterjee, A.: Regenerative tool chatter near a codimension 2 hopf point using multiple scales. Nonlinear Dyn. 40(4), 323–338 (2005). https://doi.org/10.1007/s11071-005-7292-9

    Article  MathSciNet  Google Scholar 

  37. Wang, H., Hu, H.: Remarks on the perturbation methods in solving the second-order delay differential equations. Nonlinear Dyn. 33(4), 379–398 (2003). https://doi.org/10.1023/b:nody.0000009957.42817.4f

    Article  Google Scholar 

  38. Wirkus, S., Rand, R.: The dynamics of two coupled van der pol oscillators with delay coupling. Nonlinear Dyn. 30(3), 205–221 (2002). https://doi.org/10.1023/a:1020536525009

    Article  MathSciNet  Google Scholar 

  39. Wright, E.M.: The non-linear difference-differential equation. Q. J. Math. 17(1), 245–252 (1946). https://doi.org/10.1093/qmath/os-17.1.245

  40. Zhang, J.F., Huang, F.: Nonlinear dynamics of a delayed leslie predator–prey model. Nonlinear Dyn. 77(4), 1577–1588 (2014). https://doi.org/10.1007/s11071-014-1400-7

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author would like to thank LPDP Indonesia for the scholarship of the Doctoral Program. This research was funded by the Directorate for the Higher Education, Ministry of Research, Technology, and Higher Education of Indonesia, through the Research Grant Penelitian Disertasi Doktor (PDD), Universitas Gadjah Mada 2022, no. 1739/UN1/DITLIT/Dit- Lit/PT.01.03/2022.

Funding

This research was funded by the Directorate for the Higher Education, Ministry of Research, Technology, and Higher Education of Indonesia, through the Research Grant Penelitian Disertasi Doktor (PDD), Universitas Gadjah Mada 2022, no. 1739/UN1/DITLIT/DitLit /PT.01.03/2022.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia

    N. Binatari, F. Adi-Kusumo & L. Aryati

  2. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    W. T. van Horssen & P. Verstraten

  3. Department of Mathematics Education, Faculty of Mathematics and Natural Sciences, Universitas Negeri Yogyakarta, Yogyakarta, Indonesia

    N. Binatari

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  1. N. Binatari
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Correspondence to F. Adi-Kusumo.

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N. Binatari: On leave as a doctoral student at Universitas Gadjah Mada.

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Binatari, N., van Horssen, W.T., Verstraten, P. et al. On the multiple time-scales perturbation method for differential-delay equations. Nonlinear Dyn 112, 8431–8451 (2024). https://doi.org/10.1007/s11071-024-09485-z

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